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A168532 Triangle read by rows, A054525 * A168021. 6
1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 6, 0, 0, 0, 1, 7, 2, 1, 0, 0, 1, 14, 0, 0, 0, 0, 0, 1, 17, 3, 0, 1, 0, 0, 0, 1, 27, 0, 2, 0, 0, 0, 0, 0, 1, 34, 6, 0, 0, 1, 0, 0, 0, 0, 1, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 63, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums = A000041 starting (1, 2, 3, 5, 7, 11, 15, ...).

T(n,k) is the number of partitions of n into parts with GCD = k. - Alois P. Heinz, Jun 06 2013

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

Mobius transform of triangle A168021 = an infinite lower triangular matrix with aerated variants of A000837 in each column; where A000837 = the Mobius transform of the partition numbers, A000041.

EXAMPLE

First few rows of the triangle =

1;

1,    1;

2,    0, 1;

3,    1, 0, 1;

6,    0, 0, 0,  1;

7,    2, 1, 0,  0, 1;

14,   0, 0, 0,  0, 0, 1;

17,   3, 0, 1,  0, 0, 0, 1;

27,   0, 2, 0,  0, 0, 0, 0, 1;

34,   6, 0, 0,  1, 0, 0, 0, 0, 1;

55,   0, 0, 0,  0, 0, 0, 0, 0, 0, 1;

63,   7, 3, 2,  0, 1, 0, 0, 0, 0, 0, 1;

100,  0, 0, 0,  0, 0, 0, 0, 0, 0, 0, 0, 1;

119, 14, 0, 0,  0, 0, 1, 0, 0, 0, 0, 0, 0, 1;

167,  0, 6, 0,  2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

209, 17, 0, 3,  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;

296,  0, 0, 0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

...

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, x,

      b(n, i-1)+(p-> add(coeff(p, x, t)*x^igcd(t, i),

      t=0..degree(p)))(add(b(n-i*j, i-1), j=1..n/i))))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):

seq(T(n), n=1..17);  # Alois P. Heinz, Mar 29 2015

MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, 1, If[i==1, x, b[n, i-1] + Function[{p}, Sum[Coefficient[p, x, t]*x^GCD[t, i], {t, 0, Exponent[p, x]}]][Sum[b[n - i*j, i-1], {j, 1, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 17}] // Flatten (* Jean-Fran├žois Alcover, Jan 08 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A168021, A000837.

Cf. A256067 (the same for LCM).

Sequence in context: A026794 A137712 A194711 * A181940 A261209 A093555

Adjacent sequences:  A168529 A168530 A168531 * A168533 A168534 A168535

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Nov 28 2009

EXTENSIONS

Corrected and extended by Alois P. Heinz, Jun 06 2013

STATUS

approved

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Last modified August 19 16:24 EDT 2017. Contains 290809 sequences.